Anyone who has taken a year of Calculus (including integration) will fondly remember volumes by slicing. The basic idea is that if you have a solid object and a line running through it, then the volume of the solid is the limit of the sum of all the cross sections of the solid perpendicular to the line of thickness as approaches zero. In the language of calculus, this becomes:
A(x) dx
where a and b are the points on the line at the ends of the solid and A(x) is the area of a cross section of the solid perependicular to the line at the point of coordinate x (assuming the line is outfitted with a coordinate system). The most common problems of this type are as follows:

Find the volume of the solid composed of squares perpendicular to the x-axis with the area enclosed by y=x² - 4 and the x-axis as a base.

So far, so good. The area of the square cross sections are the squares of the length of the side; that is, (4 - x²)². So the volume of the solid will be:
(4 - x²)² dx
All well and good; evaluating that integral (which is easy: it's just a polynomial function) will give is the volume of the solid in question. But there's always a nagging question in the back of the mind of the calculus student: What does that solid actually look like?
Well, wonder no more, calculus students of the world! Every year the NHHS BC calculus class creates models of such solids in styrofoam. They divide up into groups of two and each group is given one problem such as the one above. On one large piece of styrofoam they mark out the base, and them measure the width at intervals corresponding to the width of a piece of styrofoam. They then cut out the shape given of different size at each of these intervals, and glue them all together on top of the base, producing a reasonably accurate model of what such a solid actually looks like.
Hopefully in the future we will have more pictures of some of these models here for you to look at. In the meantime, you'll just have to wait.


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